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388 PRAGUE ECONOMIC PAPERS, 3, 2014

KUZNETS INVERTED U-CURVE HYPOTHESIS EXAMINED

ON UP-TO DATE OBSERVATIONS FOR 145 COUNTRIES

Oksana Melikhova, Jakub Čížek*

Abstract:

The Kuznets hypothesis of inverted U-curve dependence of the income inequality on the absolute

value of the average income is still an unresolved issue despite the growing number of theoretical

and empirical research on this topic. This paper analyzes the historical data on the average income

and the income inequality for the period 1979-2009 collected for 145 countries. We found that the

income inequality is infl uenced predominantly by governmental policy on subsidies and social

transfers. Different amount of subsidies and social transfers across various countries makes the

data biased. The inverted U-curve was found in countries with low amount of social contribution.

However, increasing amount of social contributions makes the U-curve fl at and shifts its maximum

to higher values of the average income. Based on the experimental data a model describing the

infl uence of both governmental policy and the level of economic development was developed.

Keywords: Kuznets curve, inverted U-curve, income inequality, comparative theory.

JEL Classifi cation: O15, C23

1. Introduction

Simon Smith Kuznets (1955) was the fi rst who suggested existence of a general

relationship between the income inequality and the income per capita. His hypothesis

states that the income inequality initially rises with economic development but after

reaching its maximum it subsequently falls in advanced stages of economic development.

Hence, the relationship between the income inequality and the average income expressed

as GDP per capita has shape of inverted U-curve.

The Kuznets inverted U-curve hypothesis can be undoubtedly considered as one of the

most infl uential statements ever made on inequality and development (Moran, 2005).

Since Kuznets pioneering work in 1955 a number of theories have been proposed to

explain the inverted U-curve relationship between the income inequality and the level

of economic development. Anand and Kanbur (1993a) followed the original Kuznets

idea and developed an inter-sectoral migration model. Williamson (1991) proposed that

change in technology initially increases the income inequality, but it declines when skills

* Institute of Economic Studies, Faculty of Social Sciences, and Faculty of Mathematics and Physics,

Charles University in Prague, V Holešovičkách 2, Praha 8, Czech Republic (oksana.melikhova@

mff.cuni.cz; [email protected]).

This work is supported by the Grant Agency of Charles University in Prague

(Project GA UK № 334111)

DOI: 10.18267/j.pep.490

PRAGUE ECONOMIC PAPERS, 3, 2014 389

related to the new technology become widespread. Barro (2000) suggested that each

implementation of new technology is accompanied by dynamic Kuznets-type effect on

the distribution of income. Aghion and Bolton (1997) attributed the Kuznets U-curve

to credit market imperfections which cause different behaviour among the poor and the

rich: in early stages of economical development the rich get richer, while the poor remain

poor, at later stages of economical development the accumulation of wealth by the rich

people pushes down their interest rates and allow the poor to become richer.

Although many theoretical models can predict the Kuznets U-curve, empirical evidence

for the validity of the Kuznets hypothesis is still a matter of controversy. Empirical

research on the validity of Kuznets hypothesis was performed by many authors during

last 30 years, but obtained results are controversial and not conclusive. Ahluwalia

(1976) in his early work found support for the Kuznets hypothesis. However, Anand

and Kanbur (1993a) re-analyzed later the same data as Ahluwalia (1976) and did not

fi nd any evidence supporting the Kuznets hypothesis. In addition, Anand and Kanbur

(1993b) analyzed also additional data than those used by Ahluwalia and again did not

fi nd any evidence for the Kuznets hypothesis. Deininger and Squire (1998) performed

both cross-country analysis and examination of country specifi c time series. They did not

found any support for the Kuznets inverted U-curve neither in the cross-country analysis

nor in the country specifi c inter-temporal data. On the other hand, Jha (1996) analyzed

observations for 76 countries for the period 1960-92 and found that Kuznets hypothesis

holds. Similarly Milanovic (2000) reported that Kuznets hypothesis was supported by

data for 80 countries during the 1980s. Bulir (2001) came to similar conclusion from

analysis of cross-sectional data for 75 countries. Parametric and semi-parametric testing

of Kuznets hypothesis performed by Lin et al. (2006) on the same data as Bulir (2001)

gave also support for the Kuznets hypothesis. But recently Tam (2008) analyzed ‘high

quality’ data for 84 countries and did not fi nd any support for the Kuznets hypothesis.

The income inequality is usually measured by the GINI coeffi cient while the GDP per

capita characterizes the level of economic development. Most studies followed the

approach of Ahluwalia (1976) and performed linear regression of GINI index on the

logarithm of GDP per capita and its square term, i.e. the relationship between GINI

index and GDP per capita is assumed in the form

2

0 1 2GINI log GDP log GDPi ii i . (1)

Positive coeffi cient β1 and negative β2 obtained from regression are viewed as a support

of the validness of Kuznets hypothesis. The maximum of the inverted U-curve described

by Equation (1) occurs at GDP per capita

1

22

TPGDP 10 ,

(2)

which according to the Kuznets hypothesis represents the ‘turning point’, i.e. the level

of development from which inequality should decrease with further economical deve-

lopment. The maximum value of GINI coeffi cient predicted by the Kuznets U-curve is

DOI: 10.18267/j.pep.490


2

1max 0

2

GINI .4

(3)

The both quantities GDPTP and GINImax calculated from results of analysis performed

previously by various authors are listed in Table 1.

Table 1

Summarized Results of Previous Data Analysis Testing Validity of the Kuznets Hypothesis

PeriodNumber of

countries

Number

of pairs

Fraction of

complete

data (%)

GINImax

(%)

GDPTP

(constant

2000 US$)

Reference

1965-1971 60 60 0.6 57.6 642 Ahluwalia (1976)

1970-1990 75 75 0.8 62.7 2221 Bulir (2001)

1990-2000 44 45 0.5 45.0 2575 Hayami (2005)

1965-2003 82 82 0.8 46.0 2570 Iradian (2005)

1970-1990 75 75 0.8 45.9 912 Lin et al. (2006)

1979-2008 145 630 6.3 43.8 1528 This work

Note: The columns of the table show the time period covered by analyzed data, number of countries included in the

analysis, number of samples (i.e. matching pairs of GINI index and GDP per capita), the fraction of the complete data

set (approximately 10,000 pairs) represented by number of samples used in analysis, the maximum value GINImax

of the fi tted model function and the position of the turning point GDPTP of the fi tted model function. Results of linear

regression of data in Figure 1 by Equation (1) are shown in the table for comparison.

One can see in Table 1 that GDPTP and GINImax exhibit a big scatter which hinders any

meaningful prediction of the turning point position. Since in the Kuznets hypothesis

GDPTP and GINImax are principal quantities describing a fundamental economical law,

the failure of various analyses to agree at GDPTP and GINImax arises some doubts about

the validity of the Kuznets hypothesis.

The empirical analyses of Kuznets hypothesis suffer from a small number of data

available. The income inequality is being systematically traced from 1960, i.e. for about

of 50 years. Taking into account the number of countries in the world, a complete data set

containing GINI index and GDP per capita pairs measured annually since 1960 would

contain approximately 10,000 pairs. Unfortunately only a small fraction of these data is

available mainly due to lack of information about GINI index. Table 1 summarizes the

time period, the number of countries involved in investigation and the number of GINI

index - GDP per capita pairs used in the previous studies published in literature. One

can see in Table 1 that the number of GINI index - GDP per capita pairs analyzed so far

represents less than 1% of the complete data set.

In this work we performed a detailed analysis of the relationship between the income

inequality and the level of economic development on a large sample representing

6.3% of the complete dataset and we developed a model which describes the empirical

observations.

DOI: 10.18267/j.pep.490


2. Results

Analyzed sample consists of data available in the online World Bank database World

Development Indicators (WDI) (2011) for 145 countries for which GINI index was

measured at least once in the period 1979-2009. In total WDI contains 6,228 records

about GDP per capita and 644 records about GINI index for various countries in the

period 1979-2009. There are 630 matching pairs of GINI index and GDP per capita for

the same country and the same year. Hence, our sample represents 6.3 % of the complete

data set, which is almost 8 times larger size of the sample compared to previous works.

Figure 1 shows all data as a scatter plot of GINI index against the common logarithm of

GDP per capita (expressed in constant US$ 2,000) for the same country in the same year.

Basic statistical description of all available observations and data for which matching

pairs can be found are given in Table 2. Because of the limited number of GINI and GDP

per capita pairs, it is important to check whether data in Figure 1 can be considered as

a representative sample from the complete data set. Figure 2(a) and (b) show marginal

probability density function (pdf) for GINI index and GDP per capita, respectively. The

marginal pdf constructed from all available data irrespective if they form matching pairs

or not, i.e. 644 records of GINI index and 6,228 values of GDP per capita are plotted by

dashed lines, while solid lines show marginal pdf constructed only from the data forming

matching pairs of GINI index and GDP per capita, i.e. 630 records. Since there are only

a few values of GINI index for which the corresponding GDP per capita value is not

available, shape of both marginal pdf’s in Figure 2(a) is very similar. More interesting

is comparison of marginal pdf’s for logarithm of GDP per capita. One can see in Figure

2(b) that there is a reasonable agreement between the marginal pdf constructed from all

6,228 GDP records and that created only from GDP values for which the corresponding

GINI index value is available. In comparison with the marginal pdf constructed from all

available GDP records the pdf constructed from the matching pairs only exhibit slightly

enhanced contribution from middle-income countries (log GDP per capita falling in the

range from 3 to 3.8) at the expense of the contribution of high-income countries (log

GDP per capita > 4). The student’s t-test was employed to check if there is a statistically

signifi cant difference between the mean of the marginal pdf constructed from all GDP

per capita records and that created from the matching pairs only. The p-value from the

two sided student’s t-test (assuming as a null hypothesis that both distributions have the

same mean) is P = 0.30. Hence, we can conclude that there is no statistically signifi cant

difference in the mean of both pdf’s in Figure 2(b) and the data in Figure 1 can be

considered as a representative sample.

Visual inspection of data in Figure 1 did not reveal any regular pattern which would

suggest the inverted U-curve dependence of the inequality on the logarithm of GDP per

capita. Correlation tests were employed to examine possible relationship between the

GINI index and the logarithm of GDP per capita. Results of these tests are summarized

in Table 3.

DOI: 10.18267/j.pep.490


Figure 1

Dependence of the GINI Index on the Common Logarithm of GDP per Capita for 145 Countries

Note: All historical observations in period 1979-2008 are plotted. Curved line shows result of linear regression using

the model function described by Equation (1). Straight line shows the mean GINI index calculated as arithmetic

average of all entries.

Source: WDI (2011), authors’ calculations.

Table 2

Basic Statistic Description of GINI Index, Log GDP per Capita, Social Contributions (% GDP)

and Subsidies and Other Transfers (% of GDP)

samples mean median std min max

GINI 644 42.0 41.5 10.2 19.4 74.3

log GDP per capita 6228 3.35 3.31 0.70 1.79 5.07

social contributions (% of GDP) 1220 6.26 5.29 5.32 0.00 20.25

subsidies and transfers (% of GDP) 1630 11.14 9.23 8.41 0.027 47.1

GINI with matching GDP per capita

pairs630 42.2 41.7 10.0 19.5 74.3

GINI with matching social

contributions (% of GDP) pairs242 41.2 38.86 10.6 21.6 65.8

GINI with matching subsidies and

transfers (% of GDP) pairs232 39.1 36.8 9.91 21.6 61.0

log GDP per capita with matching

GINI pairs630 3.19 3.23 0.52 1.96 4.67

social contributions (% of GDP)

with matching GINI pairs242 5.87 4.99 4.64 0.00 19.4

subsidies and transfers (% of GDP)

with matching GINI pairs232 11.15 9.84 7.43 0.169 31.9

log (GDP per capita)

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

GIN

I in

dex (

%)

10

20

30

40

50

60

70

80

fit by Eq. (1)

mean GINI index

DOI: 10.18267/j.pep.490


Figure 2

Marginal pdf of (a) the GINI Index and (b) the Logarithm of GDP per Capita

Note: Dashed lines show marginal pdf constructed from all available data of the GINI index (N = 644) and the GDP per

capita (N = 6228) irrespective if they form matching pairs or not. The solid lines show marginal pdf constructed only from

data forming matching pairs of the GINI index and the GDP per capita for the same country and the same year (N = 630).

Dash-dotted line in the upper panel shows marginal pdf of the GINI index observations for which corresponding social

contributions data forming matching pair for the same country and the same year are available (N = 242).


The Pearson sample correlation coeffi cient ρ = 0.008 was calculated for the data plotted in

Figure 1. The P-value of 0.47 determined for the Pearson correlation coeffi cient by a two

sided permutation test (assuming as a null hypothesis zero correlation between GINI and

GDP per capita) testifi es that there is no statistically signifi cant Pearson correlation. This

result should be taken with caution since the Pearson coeffi cient is a measure of a linear

log GDP per capita

2.0 2.5 3.0 3.5 4.0 4.5 5.0

frequency

0.0

0.2

0.4

0.6

0.8

1.0all data N = 6228

pairs GDP with GINI N = 630

GINI index (%)

20 30 40 50 60 70

frequency

0.00

0.01

0.02

0.03

0.04

0.05

0.06

all data N = 644

pairs GINI with GDP N = 630

pairs GINI with social contribution N = 242

(a)

(b)

DOI: 10.18267/j.pep.490


dependence between random variables, while the Kuznets hypothesis predicts quadratic

relationship between the GINI index and the logarithm of GDP per capita. Therefore,

we analysed the data in Figure 1 also using the Spearman rank order correlation test

which examines whether there is a monotonic relationship between both quantities but it

is independent on shape of this dependence. Results of this analysis shown in Table 3 do

not show any statistically signifi cant Spearman correlation.

Table 3

Results of Statistical Correlation Tests and the χ2 Statistical Test of Independence

Method

Test statistics:correlation coeffi cient / 2 per degree of freedom

95 % confi dence interval

P-value

(i) GINI index versus logarithm of GDP per capita

Pearson correlation test 0.008 (-0.086, 0.086) 0.47

Spearman correlation test 0.061 (-0.081, 0.081) 0.14

2 test of independence 0.94 (0.89, 1.11) 0.32

(ii) GINI index versus social contributions (expressed in % of GDP)

Pearson correlation test -0.620 (-0.12, 0.12) < 0.001

Spearman correlation test -0.621 (-0.10, 0.10) < 0.001

2 test of independence 3.22 (0.81, 1.19) < 0.001

(iii) GINI index versus subsidies and other transfers (expressed in % of GDP)

Pearson correlation test -0.497 (-0.12, 0.12) < 0.001

Spearman correlation test -0.504 (-0.10, 0.10) < 0.001


(iv) GINI – β s,1 si versus logarithm of GDP per capita

Pearson correlation test 0.329 (-0.12, 0.12) < 0.001

Spearman correlation test 0.371 (-0.10, 0.10) < 0.001


Note: The tests are applied on (i) pairs of the GINI index and the logarithm of GDP per capita; (ii) pairs of the GINI index

and the subsidies and other transfers (expressed in % of GDP); (iii) pairs of the GINI index and the social contributions

(expressed in % of GDP); (iv) pairs of the part of GINI index unexplained by social contributions (GINIi – βs,1 si) and

the logarithm of GDP per capita. The 95% confi dence interval for the correlation coeffi cients was determined using

permutation test. The P-value in the last column of table was calculated assuming zero correlation or independence

of investigated quantities as the null hypothesis in correlation and χ2 test, respectively.

The Kuznets hypothesis predicts inverted U-curve relation between the GINI index and

the logarithm of GDP per capita. If by chance the position of maximum of this dependence

is located close to the median of the logarithm of GDP per capita, then the resulting

correlation coeffi cients could be close to zero even if the Kuznets hypothesis holds. For

this reason we analysed data in Figure 1 also by the χ2 test of independence which can

be considered as an ultimate test for existence of any relationship between the GINI

index and the logarithm of GDP per capita. Assuming as a null hypothesis that the GINI

index and the logarithm of GDP per capita are independent variables, the χ2 test yields

DOI: 10.18267/j.pep.490


the value of χ2 per degree of freedom χ2 /ν = 0.94 and the p-value of 0.32 (see Table 3).

Hence, the χ2 test testifi es that there is no reason to reject the null hypothesis and no

support for the inverted U-curve dependence of the GINI index on the GDP per capita

was found. This result indicates that the income inequality is infl uenced predominantly

by other factors than the level of economic development.

Table 4

Results of Linear Regressions of Empirical Data

Quantity Coeffi cient Standard error P-value

(i) Kuznets curve, samples N = 630

GINIi = 0++GDP,1 (log GDP)i +GDP,2 (log GDP)i2+i

0 -14 11 0.20

GDP,1 36.3 7.2 < 0.001

GDP,2 -5.7 1.1 < 0.001

R2 (R2 adj) 0.039 (0.036)

F 16.1 P(F|H0) 2.8 10-7

(ii) Linear regression on transfer and subsidies (% of GDP), samples N = 232

GINIi = 0+t,1 ti +i

0 46.51 0.99 < 0.001

t,1 -0.637 0.074 < 0.001

R2 (R2 adj) 0.247 (0.244)

F 61.6 P(F|H0) 1.6 10-13

(iii) Linear regression on social contribution (% of GPD), samples N = 242

GINIi =0 + s,1 si + i

0 48.26 0.82 < 0.001

s,1 -1.233 0.079 < 0.001

R2 (R2 adj) 0.505 (0.503)

F 246.1 P(F|H0) 1.1 10-38

(iv) Linear regression on social contribution (% of GPD) and Kuznets curve, samples N = 242

GINIi = 0+s,1 si + GDP,1 (log GDP)i +GDP,2 (log GDP)i2+i

0 -22 12 0.056

GDP,1 38.7 7.2 < 0.001

GDP,2 -4.9 1.1 < 0.001

s,1 -1.64 0.10 < 0.001

R2 (R2 adj) 0.560 (0.555)

F 103.1 P(F|H0) 8.3 10-43

Note: The linear regressions of empirical data were made using: (i) Equation (1), i.e. dependence of the GINI index on

the logarithm of GDP per capita and its square, see Figure 1; (ii) Equation (5) i.e. dependence of the GINI index on the

volume of subsidies and other transfers ti (expressed in % of GDP), see Figure 6; (iii) Equation (4), i.e. dependence of the

GINI index on the amount of social contributions si (expressed in % of GDP), see Figure 5; (iv) Equation (6) containing a

contribution of the social transfers and also square and linear term of the logarithm of GDP per capita. Goodness-of-fi t is

characterized by the R2 coeffi cient and the R2 adjusted to the number of degrees of freedom (R2 adj) and also by F-test:

F-value and the probability P(F|H0) that the null hypothesis (H0 : all regression coeffi cients except of β0 are zero) is true.

DOI: 10.18267/j.pep.490


Table 4 shows result of linear regression of data in Figure 1 by Equation (1). The

parameters GDPTP and GINImax calculated from fi tted parameters by equations (2) and

(3) are listed in Table 1 and the model curve is plotted in Figure 1 by solid line. From

fi tting we obtained a negative value of β2 and a positive value of β1 coeffi cient which

is in accordance with the inverted U-curve hypothesis. Both coeffi cients β2 and β1 are

statistically signifi cant (at the 1% signifi cance level), which was confi rmed also by the

Fisher F-test, while uncertainty of the absolute coeffi cient β0 is very large making this

parameter statistically insignifi cant. The R2-value of 0.039 achieved in fi tting by Equation

(1) is rather low which testifi es to a poor agreement of the inverted U-shape curve with

empirical observations. Thus, GDP per capita can explain only a small portion of the

income inequality data. Similar conclusion has been drawn by Barro (2000).

Figure 1 presents aggregated data for the whole period 1979-2009. However, both the

income inequality and the GDP per capita are quantities undergoing certain development

with time. To examine this effect the mean value of GINI index and the logarithm of

GDP per capita were calculated for each year and are plotted in Figure 3 (a) and (b),

respectively. The number of records available in each year is shown in the fi gure as well.

Figure 3

Time Development of the Mean Value of (a) the GINI Index, (b) the Logarithm of GDP per Capita

Note: Error bars indicate uncertainty of the mean values expressed as one standard deviation of the arithmetic

average estimated from the spread of data in each year. Gray columns show the number of records for the GINI

index and the GDP per capita available in the WDI for each year. The solid line in the upper panel shows GINI Index

long-term average.


year

1980 1985 1990 1995 2000 2005 2010

num

ber

of

record

s

0

10

20

30

40

50

GIN

I in

dex (

%)

20

30

40

50

60

70

80

(a)

DOI: 10.18267/j.pep.490


From inspection of Figure 3 it becomes clear that the mean value of the GINI index

remains approximately constant and exhibits only statistical fl uctuations around a

long-term average indicated in Figure 3 (a) by a solid line. On the other hand, the mean

value of the logarithm of GDP per capita strongly increases in the period 1995-2007, see

Figure 3 (b). This marked increase in the GDP per capita remains, however, unnoticed

by the GINI index, which supports the picture drawn above that the GINI index is

predominantly infl uenced by other factors than the GDP per capita.

The mean value of the national GINI indexes calculated as an arithmetic average of all

data in Figure 1 is GINImean = (42 ± 1) %, see the solid line in Figure 1 and Figure 3 (a).

Banerjee and Yakovenko (2010) showed that when limited amount of money is distributed

randomly between many agents then the principle of maximum entropy tells us that under

constraints of constant total number of agents and total amount of money the equilibrium

distribution of money between the agents is given by exponentially decaying function.

Drăgulescu (2001) calculated that the GINI index for such exponential distribution equals

50 %. The mean value of the national GINI indexes is lower than 50 % corresponding

to a random distribution of resources among inhabitants in each country. This is most

probably due to active policy of governments in majority of countries with purpose to

decrease the inequality of the income distribution of their citizens. Attempts to decrease

the inequality of the income distribution are realized most frequently by social transfers

of money from the rich members of society towards the poor citizens. Hence, if this policy

works one should expect that the GINI index is infl uenced by the volume of the govern-

mental social transfers. WDI contains data about social contributions (expressed in % of

revenue). The social contributions multiplied by the revenue (expressed in % of GDP)

give a measure of resources used by government to decrease the income inequality. From

WDI we obtained 1,220 records for social contributions and these observations form

242 matching pairs with GINI for the same country and the same year (see Table 2).

Hence, this sample represents 2.4% of the complete dataset. Again it was checked

whether the matching pairs of GINI index and social contributions can be considered as a

representative sample. Dash-dotted line in Figure 2(a) shows the marginal pdf of the GINI

index records for which corresponding data about the social contributions are available.

The shape of this distribution is in very reasonable agreement with pdf constructed from

all available values of GINI index. Application of the student’s t-test did not reveal any

statistically signifi cant difference in the mean of both distributions. Comparison of the

marginal pdf of the percentage of social contributions constructed from all available data

(N = 1220, dashed line) and that constructed only from data forming matching pairs with

GINI index (N = 242, solid line) is shown in Figure 4.

DOI: 10.18267/j.pep.490


Figure 4

Marginal pdf of the Social Contributions

Note: Dashed line shows marginal pdf constructed from all available data (N = 1220) while solid line shows marginal

pdf constructed only from data forming matching pairs with the GINI index for the same country and the same year

(N = 242).


Shape of both pdf’s is very similar and the student’s t-test confi rmed that there is no

signifi cant difference between both distributions. Hence 242 matching pairs of the GINI

index and the social contribution can be considered as a representative sample.

Figure 5 shows a scatter plot of the GINI index plotted against the social contributions

(expressed in % of GDP). One can see in the fi gure that there is indeed a decreasing trend

of the income inequality with increasing fraction of social contributions. The relationship

between the GINI index and the social contributions was examined again using the

Pearson and Spearman correlation coeffi cients. Results of these tests listed in Table 3

testify that there is statistically highly signifi cant (P-value < 0.001) negative correlation

between the GINI index and the social contributions. The χ2 test of independence

employed to pairs of GINI index and the social contribution revealed that these quantities

are almost certainly (p-value < 0.001) not independent. The relation between the GINI

index and the social contributions si can be well fi tted by a linear dependence

0 1GINIi i is (4)

The coeffi cients β0, β1 obtained from heteroscedasticity corrected weighted least square

linear regression of data in Figure 5 are listed in Table 4. The linear coeffi cient β1 is

0 5 10 15 20

fre

qu

en

cy

0.00

0.05

0.10

0.15

0.20

0.25

all data N = 1220

pairs social contribution with GINI N = 242

Social contribution (% of GDP)

DOI: 10.18267/j.pep.490


negative and highly statistically signifi cant. The R2 value of 0.505 obtained in fi tting by

Equation (4) suggests that social contributions can explain a large portion of the cross-

country differences in the income inequality.

If the social contributions are zero then linear regression by Equation (4) predicts GINI

index (48.3 ± 0.8) % (see Table 4). This value is comparable with abovementioned

GINI index of 50 % corresponding to a random distribution of limited resources among

inhabitants. From the results in Table 4 one can easily calculate that fully equal income

distribution characterized by GINI = 0 is achieved when the volume of social contributions

becomes s = (39.1 ± 0.1) expressed in % of GDP. Note that such volume of social

contributions is roughly two times higher than the maximum si observation in our data set

si,max = 20.25 % corresponding to France in 1996.

Figure 5

Dependence of the GINI Index on the Social Contributions

Note: The solid line shows linear regression by Equation (4). This plot was constructed from 242 matching pairs of the

GINI index and the social contributions (expressed in % of GDP).


The volume of governmental subsidies and transfers (expressed in % GDP) can be

considered as another indicator of the government attempts to reduce the inequality of

the income distribution. This indicator was used together with the logarithm of GDP per

capita and its square term as one of the explanatory variables in linear regression of the

GINI index by Milanovic (2000). The WDI (2011) contains 1,630 records for volume

of subsidies and other transfers (expressed in % of GDP) for the period 1979-2009 (see

Table 2). From these data we can make 232 matching pairs of the GINI index and the

percentage of subsidies and other transfers representing 2.3% of the complete dataset.

0 5 10 15 20

GIN

I in

dex (

%)

10

20

30

40

50

60

70

Social contributions (% of GDP)

DOI: 10.18267/j.pep.490


Again using the t-test it was confi rmed that the matching pairs of the GINI index and the

percentage of subsidies and other transfers can be considered as a representative sample.

Figure 6 shows dependence of the GINI index on the volume of subsidies and other

transfers (expressed in % of GDP).

Figure 6

Dependence of the GINI Index on the Volume of Subsidies and Other Transfers

Note: The volume of subsidies and other transfers is expressed in % of GDP. The solid line shows linear regression

by Equation (5). This plot was created from 232 matching pairs of the GINI index and the volume of subsidies and

other transfers.


The GINI index decreases with increasing volume of subsidies and other transfers.

Statistically signifi cant negative correlation between the GINI index and the volume

of subsidies and other transfers was found using both the Pearson and the Spearman

correlation test (see Table 3). However, from inspection of Table 3 it is clear that GINI

index is less correlated with the volume of subsidies and other transfers than with the

social contributions. The dependence of GINI index on the volume of subsidies and other

transfers ti was fi tted by a linear dependence

0 1GINIi i it (5)

and results are shown in Table 4. A weighted least square method was used to account

for heteroscedasticity of data. Similarly to dependence of the GINI index on the social

contribution the linear coeffi cient β1 obtained from regression is negative and highly

statistically signifi cant. However, the R2 value for the linear fi t of the GINI index on

the volume of subsidies and other transfers by Equation (5) is signifi cantly lower

than the R2 value for the linear fi t of the GINI index on the social contributions by

0 10 20 30

GIN

I in

dex (

%)

10

20

30

40

50

60

70

Subsidies and other transfers (% of GDP)

DOI: 10.18267/j.pep.490


Equation (4). Taking into account that number of samples in fi tting by Equation (4) and

(5) was comparable one can conclude that the volume of subsidies and other transfers is

less explanatory variable than the amount of social contributions.

Figure 7 shows the volume of subsidies and other transfers plotted versus the social

contributions. As expected these two variables are strongly correlated, the Pearson

correlation coeffi cient is ρ = 0.873. However, detailed examination revealed that

subsidies and other transfers contain not only transfers intended for decreasing the

income inequality but also transfers made for other purposes. This can be well seen in

Figure 7 which contains some records with relatively large volume of subsidies and other

transfers, while the social contribution is very low or even zero. For this reason social

contributions represent better measure of government attempts to decrease inequality of

income distribution than the volume of subsidies and other transfers.

Figure 7

The Volume of Subsidies and Other Transfers Plotted versus the Social Contributions

Note: The solid line shows linear regression. This plot was created from 1,131 matching pairs of the volume of

subsidies and other transfers and the social contributions.


Figure 8 (a) shows time development of the mean value of social contributions calculated

as an arithmetic average for each year for the time period 1995-2009 for which the data

about the social contributions are available in WDI (2011). For comparison the time

dependence of the mean value of the volume of subsidies and other transfers is plotted

in Figure 8 (b). In contrast to the GDP per capita plotted in Figure 3(b) which increases

in the period 1995-2007, the social contributions remain approximately constant and

also the subsidies and other transfers exhibit only very slight variation with time. This is

0 5 10 15 20

subsid

ies a

nd tra

nsfe

rs (

% G

DP

)

0

10

20

30

40

50


Subsid

ies a

nd o

ther

transfe

rs (

% o

f G

DP

)

DOI: 10.18267/j.pep.490


consistent with behaviour of the GINI index which remains essentially constant and only

fl uctuates around a long-term average, see Figure 3(a).

Thus, our analysis confi rmed that social transfers applied by governments represent

the main factor which strongly infl uences the GINI coeffi cient. Various amounts of

social contributions in various countries make the data in Figure 1 biased and causes

large scatter of the data points in the fi gure. Thus, it is clear that the amount of social

contributions have to be included into regression of data in Figure 1 leading to equation

2

0 ,1 ,1 ,2GINI log GDP log GDPi s i GDP GDP ii is . (6)

Figure 8

Time Development of the Mean Value of (a) the Social Contributions and (b) the Subsidies and

Other Transfers

Note: The long-term averages are shown by the solid lines. Gray columns show the number of records for the social

contributions and the subsidies and other transfers available in the WDI (2011) for each year.


year

1980 1985 1990 1995 2000 2005 2010

num

ber

of

record

s0

20

40

60

80

100

so

cia

l co

ntr

ibu

tio

n (

% G

DP

)

0

5

10

15

20

year

1980 1985 1990 1995 2000 2005 2010

num

ber

of

record

s

0

10

20

30

40

50

60

su

bsid

ies a

nd

tra

nsfe

rs (

% G

DP

)

0

5

10

15

20

(a)

(b)

Socia

l contr

ibutions (

% o

f G

DP

)S

ubsid

ies a

nd o

ther

transfe

rs (

% o

f G

DP

)

DOI: 10.18267/j.pep.490


Results of heteroscedasticity corrected linear regression by Equation (6) are listed in

Table 4. Negative coeffi cient βGDP,2 and positive coeffi cient βGDP1 obtained from fi tting

(both statistically signifi cant) are in agreement with the Kuznets hypothesis. The

coeffi cient βs,1 is negative and not far from that obtained from fi tting by Equation (5).

The R2 value obtained in fi tting by Equation (6) is only slightly improved compared to

that achieved in fi tting by Equation (5) assuming only the infl uence of the amount of

social contributions. This testifi es that the income inequality is infl uenced predominantly

by the amount of social contributions.

However, results of fi tting by Equation (6) indicate that some remaining part of the

income inequality could be explained by the inverted U-curve dependence of the

inequality on the level of economic development. Figure 9 shows a scatter plot of GINIi

- βs,i si, i.e. the portion of the income inequality unexplained by the amount of social

contributes, plotted verses the logarithm of GDP per capita. Contrary to the original data

in Figure 1 the unexplained part of the income inequality in Figure 9 shows clearly the

inverted U-curve dependence. The data in Figure 9 were examined by the correlation

tests and results are listed in Table 3. Both Pearson and Spearman correlation coeffi cients

were found to be positive because the number of points on the increasing part of the

inverted U-curve is much higher than in the decreasing part. The χ2 test of independence

led to rejection of the null hypothesis which indicates that data in Figure 9 cannot be

considered as independent variables. The solid line in Figure 9 shows the best fi t of data

points by a parabolic curve.

Figure 9

Scatter Plot of GINI – βs,1s versus the Common Logarithm of GDP per Capita

Note: The GINI – βs,1s represents the part of the income inequality unexplained by the social contributions. The solid

line shows best fi t by a parabola.


log GDP per capita

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

GIN

I -

s,1

s (

%)

20

30

40

50

60

70

80

DOI: 10.18267/j.pep.490


It is a question whether the shape of the inverted U-curve is infl uenced by the amount of

social contribution, i.e. whether shape of the Kuznets curve in countries with high and

low amount of social contributions differs. In a hypothetical country with extremely high

amount of social contributions where all income is equally distributed among all citizens

the Kuznets curve cannot exist because the GINI index must be always zero independently

on the level of economic development. This suggests that increasing amount of social

contributions could make that Kuznets curve weaker. To examine this problem data in

Figure 1 were divided into four groups according to the amount of social contributions. In

Figure 10 the dependence of the GINI index on the logarithm of GDP per capita is plotted

separately for countries with various amount of the social contributions: low amount 0-5%,

Figure 10(a), intermediate amount 5-10%, Figure10(b), high amount 10-15%, Figure 10(c)

and very high amount > 15%, Figure 10(d).

Figure 10

Scatter Plots of the GINI Index versus the Logarithm of GDP per Capita for Countries with

Various Amount of Social Contributions

Note: The amount of social contributions is expressed in % of GDP: (a) low amount (< 5 %), (b) intermediate amount

(5-10 %), (c) high amount (10-15 %), (d) very high amount (> 15 %).The shadowed areas show region predicted for

countries with corresponding amount of social contributions by the model described by Equation (10).


DOI: 10.18267/j.pep.490


The dependence of the GINI index on the logarithm of GDP per capita for countries

with low and intermediate amount of social contributions exhibits the inverted U-curve

predicted by the Kuznets hypothesis, see Figure 10(a, b). With increasing amount of

social contributions the dependence of the GINI index of the logarithm of GDP per

capita becomes more and more fl at and is pushed down to the lower values of the

GINI index. The dependence of the GINI index on the logarithm of GPD per capita

for countries with high amount of social contributions shown in Figure 10(c) becomes

almost fl at and compared to the curves in Figures 10(a, b) is shifted to signifi cantly lower

values of the GINI index. Finally one can see in Figure 10(d) that the dependence of the

GINI index on the logarithm of GDP per capita for countries with very high amount of

social contributions (>15%) is completely fl at although only few points are available

here because the number of such countries is quite low.

For further analysis of the empirical data it is more appropriate to rewrite Equation (1)

in the form

2

max TP

1GINI GINI 1 log GDP log GDPi i i

w , (7)

where GDPTP is the position of the turning point, GINImax is the maximum of the Kuznets

curve and w is a measure of the width of the inverted U-curve since full width at half

maximum (FWHM) of the Kuznets parabola is FWHM = (2w)1/2. From inspection of

panels in Figure 10 it becomes clear that with increasing amount of social contributions

the parabolic U-curve becomes wider and its maximum is pushed down while the

position of its maximum is shifted towards higher values of GDP per capita. Thus, the

parameters GDPTP, GINImax and w depend on the amount of social contributions and the

Kuznets curve can be expressed as

2

i max max i TP

1GINI GINI 1 1 log GDP log GDP 1

w i

i TP i i

f sf s f s

w , (8)

where the functions fmax(s), fw(s), and fTP(s) describe the dependence of the maximum,

width and the turning point position on the amount of social transfers s, i.e. a decrease of

maximum, an increase of width and a shift of the turning point position of the inverted

U-curve. The functions fmax(s), fw(s), and fTP(s) are assumed in polynomial form

,

1

knl

k k l

l

f s a s

, (9)

where ak,l are constants and the identifi er k stands for max, w or TP. The Equation (8) can

be then expressed in the form

2

,0 0

GINI log GDP ,n

k l

i i ik l ik l

c s

(10)

where ck,l are parameters of the model which are fi tted to the empirical data. Our analysis

showed that the model function described by Equation (10) can be restricted only to

terms with n ≤ 3. Inclusion of higher-order terms does not lead to further improvement

DOI: 10.18267/j.pep.490


in agreement of the model function with empirical data and the higher-order terms

are statistically insignifi cant. Moreover, the coeffi cients c1,3 and c2,3 were found to be

statistically insignifi cant and were, therefore, fi xed at zero, i.e. c1,3 = c2,3 = 0. Results of

heteroscedacity corrected linear regression by Equation (10) are shown in Table 5.

Table 5

Results of Linear Regression of Empirical Data in Figure 1 by Model Function Described by

Equation (10)

Quantity Coeffi cient Standard error P-value

Kuznets curve with parameters the amount of social contributions, samples N = 242

2

,0 0

GINI log GDP ,n

k l

i i ik l ik l

c s

(n = 3, c1,3 = c2,3 = 0)

c0,0 -18.0 8.0 0.009

c1,0 47 14 < 0.001

c2,0 -7.9 2.4 < 0.001

c0,1 -45 10 < 0.001

c0,2 3.49 0.81 < 0.001

c0,3 0.0203 0.0036 < 0.001

c1,1 22.8 5.5 < 0.001

c2,1 -2.72 0.75 < 0.001

c1,2 -2.06 0.43 < 0.001

c2,2 0.259 0.057 < 0.001

R2 (R2 adj) 0.745 (0.735)

F 75.5 P(F|H0) 6.810-64

Fk 62.5 P(Fk |H0,k) 1.210-53

Note: The model function represents the Kuznets curve with maximum, width and turning point position modifi ed by

the amount of social contributions. Goodness-of-fi t is characterized by the R2 coeffi cient and the R2 adjusted to the

number of degrees of freedom (R2 adj) and also by F-test: F-value and the probability P(F|H0) that the null hypothesis

(H0 : all regression coeffi cients except of a constant term c0,0 are zero) is true. In order to check the signifi cance of the

terms going beyond the classical Kuznets curve an additional F-test was performed using null hypothesis H0,k that

all regression coeffi cients expect of c0,0, c1,0 and c2,0, i.e. the terms included in the classical Kuznets curve, are zero.

The Fk value obtained from the F-test and the probability P(Fk |H0,k) that null hypothesis is true are given in the last

row of the table.

All parameters ck,l used in the model are statistically signifi cant at the 1% level of

signifi cance. The R2 = 0.745 obtained from the linear regression testifi es that the model

described by that Equation (10) explains large portion of empirical data. Moreover,

F-test was employed to check the statistical signifi cance of our new model described

by Equation (10) with respect to the traditional Kuznets curve, i.e. the inverted U-curve

relationship between GINI index and logarithm of GDP per capita given by Equation (1).

The results of the F-test listed in the last row of Table 5 clearly showed that our model

describes empirical data signifi cantly better than the traditional Kuznets curve.

DOI: 10.18267/j.pep.490


Figure 11 shows a two-dimensional plot of the model curve given by Equation (10) with

the parameters ck,l obtained from linear regression of experimental data. Note that the

model function is plotted only in the region below a ‘boarder line’ si = 10 (log GDP)i − 18.5

which defi nes the upper level of the volume of social contributions affordable at given

level of economic development. There are no empirical data above this boarder line. One

can clearly see in Figure 11 that dependence of the GINI index on the logarithm of GDP

per capita has indeed shape of inverted U-curve, but with increasing amount of social

contributions the maximum of the inverted U-curve is shifted to higher values of GDP

per capita and the U-curves become more and more fl at.

Figure 11

Two Dimensional Plot of the Model Described by Equation (10)

Note: Gray intensity coded GINI index is plotted as a function of the logarithm of GDP per capita and the amount of

social contributions (expressed in % GDP).


The dependence of GDPTP , i.e. the position of maximum of the inverted U-curve

representing the turning point, is plotted in Figure 12. Increasing amount of social

contributions causes a shift of the turning point to higher GDP per capita and trajectory

of this shift exhibits S-shape when plotted versus logarithm of GDP per capita. In

countries with very high amount of social contributions the position of turning point


2.0 2.5 3.0 3.5 4.0 4.5

socia

l contr

ibution (

% G

DP

)

0

2

4

6

8

10

12

0 %

10 %

20 %

30 %

40 %

50 %

GINI

DOI: 10.18267/j.pep.490


approaches infi nity and the Kuznets U-curve becomes so fl at that it practically vanishes.

The empirical data for various countries are shown in Figure 12 in the form of a bubble

plot where size of each bubble is proportional to the GINI index value. Bubbles located

close to the position of the turning point are on average larger compared to those located

far away which is in agreement with the inverted U-curve.

Figure 12

Comparison of Turning Point Position with Empirical Data

Note: Solid line shows the position of the turning point calculated from the model described by Equation (10). Empirical

data are included in the fi gure in the form of a bubble plot. Size of each bubble is proportional to the income inequality

measured by the GINI index.



1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

socia

l contr

ibitio

n (

% G

DP

)

0

2

4

6

8

10

12

14

DOI: 10.18267/j.pep.490


Figure 13

The Maximum of the Inverted U-curve (GINImax) Calculated by Equation (10) Plotted As a

Function of the Amount of Social Contributions (expressed in % GDP)


Figure 13 shows dependence of the maximum of the inverted U-curve, GINImax, on the amount of social contributions.

GINImax monotonically decreases with increasing amount of social contributions and this together with shift of GDPTP

make the inverted U-curve fl at. The shadowed areas in Figure 10 show a band defi ned by merged model curves

calculated by Equation (10) for countries with low (< 5%), intermediate (5-10%), high (10-15%) and very high (> 15%)

amount of social contributions. The area defi ned by Equation (10) is in reasonable agreement with empirical data.

3. Conclusions

The empirical test of the Kuznets hypothesis was performed on data for 145 countries

in the period 1979-2009. It was found that the income inequality is strongly infl uenced

by the amount of social contributions which makes the panel data for various countries

biased. The inverted U-curve was found in the countries with low amount of social

contributions. With increasing amount of social contribution the inverted U-curve

fl attens, its maximum decreases and position of the turning point is shifted to higher

GDP per capita. Hence, our analysis suggests that long standing controversies regarding

the existence or non-existence of the Kuznets inverted U-curve were likely caused by the

fact that the U-curve was blurred due to various amount of social contributions which

infl uence the income inequality much more than the level of economic development.

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0 2 4 6 8 10 12

GIN

I max (

%)

0

10

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50

60


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DOI: 10.18267/j.pep.490

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